Cluster Expansions: Treating the effects of strain

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Cluster Expansions Treating the effects of strain

• Gus L. W. HartNorthern Arizona
  UniversityFlagstaff Arizona, USA

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Flagstaff ArizonaIts not hot!

• Average July high/low 28 C (82 F) and 10 C
  (50 F)Average July relative humidity 37
  (afternoon)Average yearly snowfall 2.7 m (108
  in)
• Flagstaff is one of the 10 sunniest locations in
  the US but...it is also one of the top 10
  snowiest cities

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Northern Arizona UniversityAltitude 2100 m
(7000 ft)
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PLEASE! Interrupt me Ask questions Make
comments Respond to questions
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Generalized Ising model
Formally exact but useless unless sums can be
truncated
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An IllustrationCluster expansion (Ising)
Approach Determine Z for a few structures via
DFT Use fitting to find values for J
Predict Z for new structures, check against
DFT Add new structures to your computed set of
Zs Iterate until we have a universal set of Js
? Result a robust, extremely fast, model
Hamiltonian (millions of atoms,
millions of configurations)
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Cluster Expansion for
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Coherent semi-infinite slabs
Cluster expansion predictsvanishing formation
enthalpy
Who cares?
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Precipitates!
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Have a profound effect on theproperties of
metals
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Huh? What does that have to do with coherent
semi-infinite slabs?
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Coherent semi-infinite slabs
Cluster expansion predictsvanishing formation
enthalpy
Is that correct?
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Alloy of two mismatched components
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What happens at an (coherent) interface?
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And the energy?
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• The resulting strain energy is long-range.
• What do we mean by that?
• How is it that the Ising model (cluster
  expansion) cant handle this?
• How can this be fixed?

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Superlattices...

• A superlattice is like small, semi-infinite
  slabs(What the _at_ does that mean?)
• Consider a 2n-layer superlattice AnBn
• What happens as n gets large?(Or as k gets small
  if we think in k-space?)

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Its even worse...
Strain energy is direction dependent
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Non-analyticity

• In the limit of infinite-layers (k goes to zero)
  the strain reaches a finite value...
• But that value depends on the orientation of the
  interface!
• I.e., the limit k?0 depends on the direction in
  which we take the limit

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Account for the long-range part of the energy
by adding another term
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In principle, any correct model forthe strain
can be used...
Must have the correct non-analyticity
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We can still parameterize ab-initio
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Epitaxial strains
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Epitaxial strains
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Depends on both concentration and in-plane
lattice constant
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We end up with an upside-down parabola for each
direction
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Stefan Muellers work
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Stefan Muellers work